My advice to you is to use Deku Nuts and roll like crazy when going up Death Mountain. I've done a similar challenge before (in which it was 3 Hearts, No Shield for benefit, no Bottled Items, etc. etc.) and trust me, if you use Deku Nuts on Stalfos, that will make things loads easier. Although that's assuming you're doing an exact replica of my challenge, which you probably aren't.
Also, you may want to get the Biggoron Sword so that way the use of your shield is limited. And the Bow/Slingshot/Boomerang/Hookshot works on Deku Scrubs, you just gotta keep at it. Also, whatever you do don't get hit too often if you don't have milk or anything. A lot of the enemies late in the game do quite a bit of damage for a 3 Heart Run, and with no Shield, that only makes things harder.
My final piece of advice is to believe in yourself and keep on truckin'. Never give in, aim for the stars, the sky is your oyster (that didn't come out right, lol), etc. etc.